GENDER has entered the increasingly testy debate between Public Advocate Betsy Gotbaum and Mayor Bloomberg.

At a press conference this week to denounce Bloomberg’s attempt to cut the public advocate out of the line of succession for City Hall, a Gotbaum ally noted that the mayor was undercutting the city’s top female elected official.

“If he [Bloomberg] were to leave office – and that’s what we’re talking about now – we would have the first woman mayor of the City of New York,” said City Councilman Oliver Koppell (D-Bronx).

“We have a woman as the public advocate. He’s proposing a change in succession that would deny that. We can’t ignore that.”

In an interview with The Chief, the civil service weekly, Richard Steier reports that Gotbaum “dance[d] around the issue about whether Mr. Bloomberg has treated her differently because she’s a woman.”

Gotbaum went on to say that she’s known the mayor for years, compared to her fellow colleagues in city government, Comptroller Bill Thompson and City Council Speaker Gifford Miller.

But despite her long friendship with the mayor, Gotbaum pointed out: “He certainly hasn’t asked me to dinner or to play golf the way he has with those guys.”

She later told The Post she wasn’t accusing Bloomberg of sexism.

“I’m not personalizing this in any way,” Gotbaum said.

Bill Cunningham, the mayor’s communications director, said anyone raising the specter of sexism in the referendum discussion was “grasping at straws” to duck the succession issue

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The Rev. Al Sharpton’s attempt to derail new Schools Chancellor Joel Klein has flopped in record time.

Former Bronx Borough President Fernando Ferrer, endorsed by Sharpton in last year’s mayoral race, said he’s prepared to give Klein the benefit of the doubt.

“I’d like to give this man an opportunity to succeed rather than mortgage him down with all kinds of weight,” Ferrer said, 24 hours after Sharpton announced he’d try to block Klein.

Thompson, the city’s top black elected official, made it clear Klein wouldn’t have been his first choice.

At the same time, Thompson said he wasn’t about to “disagree and quibble” because the mayor has assumed control of the schools, and everyone’s “prime focus” should be “improvement for children.”

“Would it have been a different person, that might have been nice,” said Thompson. “It becomes less of an issue than it might have been under a different structure because the mayor now is ultimately in charge.”